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Message: <blockquote data-quote="EzekielRaiden" data-source="post: 8543312" data-attributes="member: 6790260">Brief digression on trig then, if it's of interest.[SPOILER="Why trig is a Thing & some of what it's used for"]At least with the trig functions, each of them actually does relate to some particular component of the angles and measures around a circle, with the actual "trigono" part of "trigonometry" being kind of secondary (but it's a lot easier to start with triangles than to start with the unit circle). Sine, as a trigonometric function of angles rather than as a ratio of right-angle triangle bits, takes an angle (by convention, an angle from the positive X axis going CCW) as its input and gives a distance as its output. In specific, if you were looking at a unit circle, it would tell you the vertical distance that you have to travel to go from the center of that circle to the point where a ray with that angle touches the circle. If you're not looking at a unit circle, it instead tells you the proportion of the circle's radius that you have to go up or down (up being positive, down being negative). Likewise, cosine takes in angles, and spits out horizontal distances (left being negative, right being positive). The ratios of right-angle triangle bits are, in essence, a stepping stone, getting the foot in the door until we can graduate to the more complete story, which is trigonometric functions on angles of any measure, even if they are too big to fit into a triangle (or are negative!)Now, this might sound...kinda pointless. Who cares what the horizontal or vertical distance proportions are? But it turns out that knowing "how much of this arrow points toward the left (etc.)?" is actually very, VERY important for physics, because it turns out that forces that work perpendicular to the direction of action (motion, electromagnetic force, gravity, etc.) do zero work. So, for example, if you know that a person is pushing against a mine cart at an angle of 45 degrees to the actual left/right track direction, you can calculate that the actual work they're doing to push the cart forward is only cos(45) = 0.707... (in exact form, sqrt(2)/2) of the amount they're actually pushing. Say they're pushing with a force of 100N (easily achievable by most humans), the cart only experiences ~70.7N of forward force; the remaining ~29.3N are wasted, pushing against the normal force (structural integrity, in this case) of the track. Cosine and sine are also essential for certain ways of multiplying vectors together, which are how we can calculate things like "the amount of work done" (requires cosine) and "if you use a coil of wire to make an electromagnet, which direction does its magnet field point?" (requires sine).[/SPOILER]Yeah, I can understand that. The calculus stuff mostly helps to skip over explaining certain things you do to functions that are extremely important but a bit of a pain to spell out all at once (mostly what a derivative is, particularly with functions of more than one variable, and why on earth anyone would care about derivatives, especially taking a derivative more than once.)</blockquote>

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